First-order linear differential equation
Contents
Definition
Format of the differential equation
A first-order linear differential equation is a differential equation of the form:
where are known functions.
Solution method and formula: indefinite integral version
Let be an antiderivative for
, so that
. Then, we multiply both sides by
. Simplifying, we get:
Integrating, we get:
Rearranging, we get:
where
is an antiderivative of
.
In particular, we obtain that:
The function is termed the integrating factor for the differential equation because multiplying by this turns the differential equation into an exact differential equation, i.e., a differential equation to which we can apply integration on both sides.
Solution method and formula: definite integral version
Suppose we are given the initial value condition that at .
Let be an antiderivative for
, so that
. Then, we multiply both sides by
. Simplifying, we get:
Integrating from to (arbitrary)
, we get:
Thus, the general expression is:
Examples
Simple example
Consider the differential equation:
Here, . Take
and get:
This gives:
Example that is better solved by subtitution
Consider:
Divide both sides by to get:
This is linear, with ,
. Take
and
(see note):
This gives:
The linear method is unnecessary -- we divided and multiplied by . A better solution would be to substitute
and get a separable differential equation.
Example where a particular solution is obtained by inspection
Consider:
The linear method gives:
The integration is not easy. So, instead of trying to do the integration directly, we note that the answer is:
It thus suffices to find a particular solution. Inspection and guesswork gives a solution . The general solution is thus: